= 2-summing norm of a finite-dimensional identity
{title2=$\pi_2(I_E)=\sqrt{\dim E}$}
For a finite-dimensional <normed vector space> $E$, represent a finite family by its column operator $A:\ell_2^m\to E$. Its operator norm is the family's weak square norm. Since $A=AQ$ for the <orthogonal projection> onto $(\ker A)^\perp$, summing $\|Ae_j\|^2\le\|A\|^2\|Qe_j\|_2^2$ proves the upper bound $\sqrt{\dim E}$. In coordinates adapted to a maximal inscribed Euclidean or Hermitian <ellipsoid>, the <John contact decomposition> gives $\sum_r c_ru_ru_r^*=I$ and $\sum_rc_r=\dim E$. The vectors $\sqrt{c_r}u_r$ have total squared norm $\dim E$ and weak square norm one, giving equality. This holds over both real and complex scalars.
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